Dual medial axes in finite-dimensional normed spaces
Abstract
We study medial axes of nonempty closed sets in finite-dimensional real normed spaces whose norms are C1 away from the origin. Using the differential of the norm, we define the dual k-medial axis by requiring the contact covectors associated with nearest points to have affine dimension at least k−1. In dimension n, we prove that this set is (n−k+1)-rectifiable for every 1≤ k≤ n+1, and hence has Hausdorff dimension at most n−k+1. We give a direct geometric proof and characterize the dual medial axes through the superdifferential of the squared distance function, obtaining an alternative proof from a theorem of Alberti, Ambrosio, and Cannarsa on singularities of semiconvex functions. Under the additional assumption of strict convexity, the ordinary and dual medial axes coincide for k=2,3, yielding the corresponding rectifiability results for ordinary medial axes. For k=4, however, the ordinary rectifiability conclusion can fail even for norms arbitrarily close to Euclidean: we construct norms on ℝ3 with smooth positively curved unit spheres arbitrarily C∞-close to the Euclidean sphere, together with four-point sets whose ordinary $4$-medial axes contain nondegenerate line segments.
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BibTeX
@misc{durand2026dual,
title = {{Dual medial axes in finite-dimensional normed spaces}},
author = {Arnaud Durand and Xiangyu Liang and Sicheng Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30878},
eprint = {2610.30878},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30878},
}
RIS
TY - UNPB
TI - Dual medial axes in finite-dimensional normed spaces
AU - Arnaud Durand
AU - Xiangyu Liang
AU - Sicheng Zhang
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30878
UR - https://arxiv.org/abs/2610.30878
AB - We study medial axes of nonempty closed sets in finite-dimensional real normed spaces whose norms are C^1 away from the origin. Using the differential of the norm, we define the dual k-medial axis by requiring the contact covectors associated with nearest points to have affine dimension at least k−1. In dimension n, we prove that this set is (n−k+1)-rectifiable for every 1≤ k≤ n+1, and hence has Hausdorff dimension at most n−k+1. We give a direct geometric proof and characterize the dual medial axes through the superdifferential of the squared distance function, obtaining an alternative proof from a theorem of Alberti, Ambrosio, and Cannarsa on singularities of semiconvex functions. Under the additional assumption of strict convexity, the ordinary and dual medial axes coincide for k=2,3, yielding the corresponding rectifiability results for ordinary medial axes. For k=4, however, the ordinary rectifiability conclusion can fail even for norms arbitrarily close to Euclidean: we construct norms on ℝ^3 with smooth positively curved unit spheres arbitrarily C^∞-close to the Euclidean sphere, together with four-point sets whose ordinary $4$-medial axes contain nondegenerate line segments.
ER -